cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A064756 a(n) = n*10^n - 1.

Original entry on oeis.org

9, 199, 2999, 39999, 499999, 5999999, 69999999, 799999999, 8999999999, 99999999999, 1099999999999, 11999999999999, 129999999999999, 1399999999999999, 14999999999999999, 159999999999999999, 1699999999999999999, 17999999999999999999, 189999999999999999999, 1999999999999999999999
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

Cf. for a(n) = n*k^n - 1: -A000012 (k=0), A001477 (k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), this sequence (k=10), A064757 (k=11), A064758 (k=12).
Cf. A064748, A126431.

Programs

  • Magma
    [ n*10^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011
  • Maple
    k:= 10; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021
  • Mathematica
    Array[# 10^# - 1 &, 18] (* Michael De Vlieger, Jan 14 2020 *)

Formula

From Elmo R. Oliveira, Sep 07 2024: (Start)
G.f.: x*(100*x^2 - 10*x - 9)/((x - 1)*(10*x - 1)^2).
E.g.f.: 1 + exp(x)*(10*x*exp(9*x) - 1).
a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3) for n > 3.
a(n) = A126431(n) - 1 = A064748(n) - 2. (End)

A064757 a(n) = n*11^n - 1.

Original entry on oeis.org

10, 241, 3992, 58563, 805254, 10629365, 136410196, 1714871047, 21221529218, 259374246009, 3138428376720, 37661140520651, 448795257871102, 5316497670165373, 62658722541234764, 735195677817154575, 8592599484487994106, 100078511642860166657, 1162022718519876379528
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Comments

Conjecture: satisfies a linear recurrence having signature (23,-143,121). - Harvey P. Dale, May 12 2019
This conjecture is true since a(n) - a(n-1) yields the recurrence 1 + 10*n + 11*n*a(n-1) - (n-1)*a(n) = 0 with polynomial coefficients in n. - Georg Fischer, Feb 19 2021

Links

Crossrefs

Cf. for a(n) = n*k^n - 1: -A000012(k=0), A001477(k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), this sequence (k=11), A064758 (k=12).
Cf. A064749.

Programs

  • Magma
    [n*11^n - 1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011
  • Maple
    k:= 11; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021
  • Mathematica
    Table[n*11^n-1,{n,20}] (* Harvey P. Dale, May 12 2019 *)

Formula

From Elmo R. Oliveira, Sep 07 2024: (Start)
G.f.: x*(121*x^2 - 11*x - 10)/((x - 1)*(11*x - 1)^2).
E.g.f.: 1 + exp(x)*(11*x*exp(10*x) - 1).
a(n) = 23*a(n-1) - 143*a(n-2) + 121*a(n-3) for n > 3.
a(n) = A064749(n) - 2. (End)

A064758 a(n) = n*12^n - 1.

Original entry on oeis.org

11, 287, 5183, 82943, 1244159, 17915903, 250822655, 3439853567, 46438023167, 619173642239, 8173092077567, 106993205379071, 1390911669927935, 17974858503684095, 231105323618795519, 2958148142320582655, 37716388814587428863, 479219999055934390271, 6070119988041835610111, 76675199848949502443519
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

Cf. for a(n) = n*k^n - 1: -A000012(k=0), A001477(k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), A064757 (k=11), this sequence (k=12).
Cf. A064750.

Programs

  • Magma
    [n*12^n - 1: n in [1..30]]; // Vincenzo Librandi, Jun 21 2018
  • Mathematica
    CoefficientList[Series[(11 + 12 x - 144 x^2) / ((1 - 12 x)^2 (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 21 2018 *)
  • PARI
    a(n) = { n*12^n - 1 } \\ Harry J. Smith, Sep 24 2009

Formula

G.f.: x*(11 + 12*x - 144*x^2)/((1 - 12*x)^2*(1 - x)). - Vincenzo Librandi, Jun 21 2018
From Elmo R. Oliveira, Sep 07 2024: (Start)
E.g.f.: 1 + exp(x)*(12*x*exp(11*x) - 1).
a(n) = 25*a(n-1) - 168*a(n-2) + 144*a(n-3) for n > 3.
a(n) = A064750(n) - 2. (End)

A242338 Numbers k such that k*7^k-1 is semiprime.

Original entry on oeis.org

1, 5, 12, 21, 42, 50, 60, 242, 272
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are: 6, 84034, 166095446411, 11729463145748964146, 13102886255950779594655873516522994057, ...
From Robert Israel, Aug 19 2014: (Start)
If k is odd, k is in the sequence iff (k*7^k-1)/2 is prime.
If k == 1 (mod 3), k is in the sequence iff (k*7^k-1)/3 is prime.
a(10) >= 506. 506*7^506 - 1 is a 431-digit composite which apparently has not been factored.
(End)

Links

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A001358, A064753, A242200.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..80] | IsSemiprime(s) where s is n*7^n-1];
  • Maple
    issemiprime:= proc(n) local F,t;
    F:= ifactors(n,easy)[2];
    t:= add(f[2],f=F);
    if t = 1 then
       if type(F[1][1],integer) then return false fi
    elif t = 2 then
       return not hastype(F,name)
    else # t > 2
       return false
    fi;
    F:= ifactors(n)[2];
    return evalb(add(f[2],f=F)=2);
end proc:
select(n -> `if`(n::odd, isprime((n*7^n-1)/2),
              issemiprime(n*7^n-1)), [$1..100]); # Robert Israel, Aug 19 2014
  • Mathematica
    Select[Range[80], PrimeOmega[# 7^# - 1]==2&]
  • PARI
    for(n=1,100,if(bigomega(n*7^n-1)==2,print1(n,", "))) \\ Derek Orr, Aug 20 2014

Extensions

a(1) = 1 prepended and comment amended by Harvey P. Dale, Aug 12 2014
a(8) and a(9) from Robert Israel, Aug 20 2014
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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